y^2+y=7

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Solution for y^2+y=7 equation: 


Simplifying
y2 + y = 7

Reorder the terms:
y + y2 = 7

Solving
y + y2 = 7

Solving for variable 'y'.

Reorder the terms:
-7 + y + y2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + y + 7 + y2 = 0 + 7

Reorder the terms:
-7 + 7 + y + y2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + y + y2 = 0 + 7
y + y2 = 0 + 7

Combine like terms: 0 + 7 = 7
y + y2 = 7

The y term is y.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
y + 0.25 + y2 = 7 + 0.25

Reorder the terms:
0.25 + y + y2 = 7 + 0.25

Combine like terms: 7 + 0.25 = 7.25
0.25 + y + y2 = 7.25

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(y + 0.5) equal to 2.692582404 and -2.692582404.


Subproblem 1
y + 0.5 = 2.692582404

Simplifying
y + 0.5 = 2.692582404

Reorder the terms:
0.5 + y = 2.692582404

Solving
0.5 + y = 2.692582404

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '-0.5' to each side of the equation.
0.5 + -0.5 + y = 2.692582404 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = 2.692582404 + -0.5
y = 2.692582404 + -0.5

Combine like terms: 2.692582404 + -0.5 = 2.192582404
y = 2.192582404

Simplifying
y = 2.192582404


Subproblem 2
y + 0.5 = -2.692582404

Simplifying
y + 0.5 = -2.692582404

Reorder the terms:
0.5 + y = -2.692582404

Solving
0.5 + y = -2.692582404

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '-0.5' to each side of the equation.
0.5 + -0.5 + y = -2.692582404 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = -2.692582404 + -0.5
y = -2.692582404 + -0.5

Combine like terms: -2.692582404 + -0.5 = -3.192582404
y = -3.192582404

Simplifying
y = -3.192582404


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.192582404, -3.192582404}

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